Scilab Function
Last update : 15/1/2004

pca - Principal components analysis

### Calling Sequence

  `[lambda,facpr,comprinc] = pca(x,N)`

### Parameters

- `x`: is a nxp (n individuals, p variables) real matrix
- `N` : is a 2x1 integer vector. Its coefficients point to the eigenvectors corresponding to the eigenvalues of the correlation matrix pxp ordered by decreasing values of eigenvalues. If N is missing, we suppose N=[1 2].
- `lambda`: is a px2 numerical matrix. In the first column we find the eigenvalues of V, where V is the correlation pxp matrix and in the second column are the ratios of the corresponding eigenvalue over the sum of eigenvalues.
- `facpr` : are the principal factors: eigenvectors of V. Each column is an eigenvector element of the dual of R^p.
- `comprinc`: are the principal components. Each column (c_i=Xu_i) of this nxn matrix is the M-orthogonal projection of individuals onto principal axis. Each one of this columns is a linear combination of the variables x1, ...,xp with maximum variance under condition u'_iM^(-1)u_i=1

### Description

This function performs several computations known as "principal component analysis". It includes drawing of "correlations circle", i.e. in the horizontal axis the correlation values r(c1;xj) and in the vertical r(c2;xj). It is an extension of the pca function.

The idea behind this method is to represent in an approximative manner a cluster of n individuals in a smaller dimensional subspace. In order to do that, it projects the cluster onto a subspace. The choice of the k-dimensional projection subspace is made in such a way that the distances in the projection have a minimal deformation: we are looking for a k-dimensional subspace such that the squares of the distances in the projection is as big as possible (in fact in a projection, distances can only stretch). In other words, inertia of the projection onto the k dimensional subspace must be maximal.

### Author

Carlos Klimann

### Bibliography

Saporta, Gilbert, Probabilites, Analyse des Donnees et Statistique, Editions Technip, Paris, 1990.